Linear-Cost Covariance Functions for Gaussian Random Fields

Abstract

Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimension, the hierarchical structure is proved to be efficient for a variety of random field computations, including sampling, kriging, and likelihood evaluation. Specifically, with n scattered sites, sampling and likelihood evaluation has an O(n) cost and kriging has an $O(\hspace{0.17em}\text{log}\hspace{0.17em}n)$ cost after preprocessing, particularly favorable for the kriging of an extremely large number of sites (e.g., predicting on more sites than observed). We demonstrate comprehensive numerical experiments to show the use of the constructed covariance functions and their appealing computation time. Numerical examples on a laptop include simulated data of size up to one million, as well as a climate data product with over two million observations.

Keywords:

• Climate data
• Gaussian sampling
• Hierarchical matrix
• Kriging
• Maximum likelihood estimation

Supplementary Material

Supplementary material includes proofs of all theorems, descriptions and pseudocode of all algorithms, and a detailed cost analysis. The software library may be accessed at https://github.com/jiechenjiechen/RLCM.

Acknowledgments

We thank anonymous reviewers whose suggestions helped improve this article significantly.

Notes

1 These libraries are the elementary components of commonly used software such as R, Matlab, and python.

2 Regression often assumes a noise term that we omit here for simplicity. An alternative way to view the noise term is that the covariance function has a nugget.

3 https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/climate-forecast-system-version2-cfsv2